Question: All of the 5th grade teachers and students from Oak went on a field trip to an archaeology museum. Tickets were $$8.00$ each for teachers and $$3.50$ each for students, and the group paid $$53.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$16.00$ each for teachers and $$11.50$ each for students, and the group paid $$133.00$ in total. Find the number of teachers and students on the field trips.
Answer: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8x+3.5y = 53}$ ${16x+11.5y = 133}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-16x-7y = -106}$ ${16x+11.5y = 133}$ Add the top and bottom equations together. $ 4.5y = 27 $ $ y = \dfrac{27}{4.5}$ ${y = 6}$ Now that you know ${y = 6}$ , plug it back into $ {8x+3.5y = 53}$ to find $x$ ${8x + 3.5}{(6)}{= 53}$ $8x+21 = 53$ $8x = 32$ $x = \dfrac{32}{8}$ ${x = 4}$ You can also plug ${y = 6}$ into $ {16x+11.5y = 133}$ and get the same answer for $x$ ${16x + 11.5}{(6)}{= 133}$ ${x = 4}$ There were $4$ teachers and $6$ students on the field trips.